Prove that $\liminf (-a_n) = -\limsup (a_n)$.

Question

Prove that $\liminf (-a_n) = -\limsup (a_n)$.

by $\liminf$ and $\limsup$ I mean the limit supremum and limit infimum of a sequence $a_n$.

do you guys have any hints?

Answer 1

I assume that the sequence $(a_n)$ is bounded. You want to show $$ \sup_m\inf_{n\geq m}(-a_n)=-\inf_m\sup_{n\geq m}(a_n)\tag{*} $$

Show first that for any bounded (real) sequence $(b_n)$: $$ \sup_n (-b_n)=-\inf_n b_n\tag{**} $$ Now use (**) to show (*).